1. Field of the Invention
The present invention relates to a determining method of movement sequence and an alignment apparatus, for example, for reducing the time of alignment between a pattern of an original plate and marks on a substrate in exposure apparatus, to a designing method and apparatus of an optical system such as a projection optical system of the exposure apparatus or a lens system for camera, and to a medium in which a program for realizing the designing method is recorded.
2. Related Background Art
In general, the exposure apparatus is arranged in such a way that, before carrying out exposure of the second layer or a layer thereafter into chip areas (or shot areas) on a wafer (photosensitive substrate) in which predetermined circuit patterns are to be formed, alignment is accomplished between a pattern of an original plate for the second or subsequent layer and the chip areas by use of EGA (a statistical arithmetic method). The EGA (Enhanced Global Alignment) is a technique for measuring positions of alignment marks (measured areas) provided mainly in the peripheral area of a plurality of selected chip areas to obtain a residual rotation error of the wafer, linear expansion or contraction of the wafer, an offset of the wafer, etc. and, based thereon, aligning all the chip areas of the wafer, for example, as disclosed in Japanese Laid-open Patent Application No. Sho 61-44429. As another technique of a further development of the EGA, U.S. Pat. No. 4,780,617 discloses an alignment technique for obtaining a residual rotation error of each chip area itself, an orthogonality error of the chip areas, and linear expansion or contraction of each chip area itself and performing alignment so as to minimize even these errors.
Particularly, when the technique of Japanese Laid-open Patent Application No. Hei 6-275496 is applied, because of the many alignment marks to be measured, the measurement time will be very long unless the alignment marks are measured as efficient as possible. For example, let us consider an example in which there are 76 exposed chip areas (areas indicated by number 01 to number 76 in the figure) in the first layer of the wafer W and four alignment marks are provided for each of the chip areas, as shown in FIG. 1. In this case, in the alignment of wafer by the EGA, the operator first selects a plurality of chip areas that are inside the outermost region and at vertices of polygon (for example, twenty chip areas hatched in FIG. 2), on an empirical basis. Coordinates of the designed center of each chip area (representing the position of each chip area) are stored in a memory of a main control system. Positions of four alignment marks of each chip area (defined by coordinates of the center thereof) are also stored in the memory of the main control system. Accordingly, the exposure apparatus was arranged to measure the position of each alignment mark according to the following movement sequence empirically seeming best, by executing the EGA.
Specifically, for example, when a measuring point of an alignment optical system is at a start point ST (x: 186.5, y: 155.5), an XY stage with a wafer mounted thereon moves so that a right upper alignment mark of a chip area closest to the start point ST (the chip area 64 in FIG. 2) comes to the measuring point of the alignment optical system (so as to be in the measuring area). After completion of the position measurement of the alignment mark, the XY stage moves so as to measure the positions of the four alignment marks counterclockwise. Next, the XY stage moves so as to measure coordinates of the right upper alignment mark of a chip area closest clockwise (the chip area 63 in FIG. 1). After that, the XY stage moves so as to measure coordinates of the four alignment marks counterclockwise. Repeating this operation, the XY stage moves to measure the positions of the alignment marks of the all chip areas selected and return the measuring point of the alignment optical system to the end point EN (x: 215, y: 133). Of course, such controls of movement were also employed that the XY stage moved so as to measure the positions of the alignment marks of each chip area clockwise and that after completion of the position measurement of the all alignment marks of a chip area, the XY stage moved so as to measure the alignment marks of a chip area closest counterclockwise.
However, the movement sequence of the XY stage in the position measurement of each alignment mark was determined empirically as described above, and no consideration was given to efficient movement control of the XY stage for the position measurement of each alignment mark.
The reason is that there arises the following problem in obtaining the movement sequence of the XY stage for the position measurement of alignment marks using the statistical measurement process such as the EGA. For example, where there are n alignment marks to be measured on the wafer, the number of conceivable stage movements for movement between alignment marks is at most nP2=n(n−1) (even though the turnaround time differs depending upon the positive or negative movement direction of the stage) and computation thereof can be done quickly. Therefore, the overall turnaround time is determined uniquely as soon as the measurement process order is determined. However, there are n! way as to the order for the measurement process of n alignment marks, and the computation time becomes too long when the all possible solutions are computed using the producing and checking method of the all conceivable orders. Particularly, if n>13, the computation is practically impossible (“Practical Course: Invitation to Traveling-Salesman Problems I, II, III,” Operations Research 39 (1994), No. 1: pp 25-31, No. 2: pp 91-96, No. 3: pp 156-162). Accordingly, the conventional alignment methods did not involve a step of finding the optimum movement sequence under practical operation conditions.
Now, let us focus attention on the optical system such as the projection optical system of the aforementioned exposure apparatus. The designing of the optical system including lens elements has been known heretofore and is known as a very difficult issue. This is because various factors, such as multiple dimensions, a super-multimodal property, strong dependent relation between variables, or complex constraints, make the issue tough. In addition, as criteria for evaluation of the optical system to be designed there exist numerous evaluation criteria such as the Seidel's five aberrations, the size, or the cost.
In the conventional designing method of optical system the basic search is a local search in the neighborhood of an initial or starting solution. If the initial solution is not appropriate, the result will fall into a local solution, so that the search will end unsuccessfully. It was thus the conventional practice to employ a method for changing the initial solution in a trial and error manner in order to find an optical system having the aimed performance. Since the conventional search basically allowed optimization of only one evaluation criterion, the designing process was changed to a single-objective process by setting a tradeoff ratio, in spite of the many evaluation criteria (Yoshiya Matsui: Lens Designing Method, Kyoritsu shuppan (1986); Jihei Nakagawa: Lens Design Engineering, Tokai daigaku shuppankai (1986); Toru Kusakawa: Lens Optics, Tokai daigaku shuppankai (1988)).
It is not possible to preliminarily know the tradeoff ratio for obtaining the optical system having the aimed performance. It is thus the present status that loads on experts are very heavy in the search for the initial solution for local search and in the search for the tradeoff ratio between the evaluation criteria.
Further, the conventional method for modifying the optical system is a method for, with data of one optical system preliminarily given as initial data by the designer, altering plural parameters, including radii of curvatures of boundary surfaces in respective optical elements (lens elements, reflectors, etc.) belonging to this optical system, distances between the boundary surfaces, and refractive indices of spaces (the lens elements and aerial lenses between the lens elements) located between the boundary surfaces, using an index of increase or decrease of a performance function indicating the performance of the lens optical system at that time.
Then the same improving procedure is repeated using the data of the optical system represented by the plural parameters after the alteration, as a new solution (i.e., the optical system to be improved). For example, if good or bad optical performance is reflected to increase or decrease of the performance function, the plural parameters will be altered so as to increase the performance function and updated each to the parameters after the alteration, as values of new parameters.
On the other hand, the genetic algorithm (GA) is known as one of optimization techniques, which imitates the evolutionary process of organism on an engineering basis. This genetic algorithm (hereinafter referred to as GA) is a generate and test method, which is characterized in that the essential point is only that dominance can be evaluated between two solution candidates. Therefore, it does not require the condition of differentiability of the performance function or the like and is thus effective to problems with complex constraints. The GA also has the feature of performing a search using a population of plural solution candidates and is drawing attention as a global search technique. Further, the GA is also drawing attention as a multi-objective optimization technique for handling the plural evaluation criteria explicitly and finding a Pareto optimal solution set by a single search.
For example, M. WALK AND J. NIKLAUS, “Some Remarks on Computer-Aided Design of Optical Lens System” (JOURNAL OF OPTIMIZATION THEORY AND APPLICATION: Vol. 59, No. 2, pp. 173-181, NOVEMBER 1988) and X. CHEN AND K. YAMAMOTO, “Genetic algorism and its application in lens design” (SPIE, Vol. 2863, PP. 216-221) describe the technology of application of the above GA to the design of optical system.